Gradient
The gradient operator is defined by , so that . Its direction is the direction of maximum increase of , its magnitude is the maximum rate of change, and is perpendicular to the contours of .
Proposition 9.2 (Properties Of Gradient Vector)
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The direction of is the direction of maximum increase of .
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The magnitude of is the maximum rate of change of , i.e.
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If is parallel to contours of , then
Therefore, is perpendicular to the contours of .
Directional derivatives
Definition 9.1 (Directional Derivative)
The directional derivative of in the direction of the unit vector is defined as
where is the angle between and .
This is the rate of change of in the direction of .
Related
Stated in
- Proposition 9.2 (Properties Of Gradient Vector)§9.1 Directional derivatives
- Definition 9.1 (Directional Derivative)§9.1 Directional derivatives